Positive mass theorem for the Yamabe problem onspin manifolds

Bernd Ammann, Emmanuel Humbert

March 15, 2003

AbstractLet (M, g) be a compact connected spin manifold of dimension n 3 whose

Yamabe invariant is positive. We assume that (M, g) is locally conformallyflat or that n {3, 4, 5}. According to a positive mass theorem by Schoen andYau the constant term in the asymptotic development of the Greens functionof the conformal Laplacian is positive if (M, g) is not conformally equivalentto the sphere. The proof was simplified by Witten with the help of spinors. Inour article we will give a proof which is even considerably shorter. Our proofis a modification of Wittens argument, but no analysis on asymptotically flatspaces is needed.

Mathematics Classification: 53C21 (Primary), 58E11, 53C27 (Secondary)

1 IntroductionThe positive mass conjecture is a famous and difficult problem which originated inphysics. The mass is a Riemannian invariant of an asymptotically flat manifold ofdimension n 3 and of order > n22 . The problem consists of proving that themass is positive if the manifold is not conformally diffeomorphic to (Rn, can). Twogood references on this subject are [LP87, Her98].

Schoen and Yau [Sch89, SY79] gave a proof if the dimension is at most 7 andWitten [Wit81, Bar86] proved the result if the manifold is spin. The positivity ofthe mass has been proved in several other particular cases (see e.g. [Sch84]), butthe conjecture in its full generality still remains open.

This problem played an important role in geometry because its solution led tothe solution of the Yamabe problem. Namely, let (M, g) be a compact connectedRiemannian manifold of dimension n 3. In [Yam60] Yamabe attempted to showthat there is a metric g conformal to g such that the scalar curvature Scalg of gis constant. However, Trudinger realized that Yamabes proof contained a seriousgap. It was the achievement of many mathematicians to finally solve the problemof finding a conformal metric g with constant scalar curvature. The problem offinding a conformal g with constant scalar curvature is called the Yamabe problem.As a first step, Trudinger [Tru68] was able to repair the gap if a conformal invariantnamed the Yamabe invariant is non-positive. The problem is much more difficultif the Yamabe invariant is positive, which is equivalent to the existence of a metricof positive scalar curvature in the conformal class of g. Aubin [Aub76] solved theproblem when n 6 and M is not locally conformally flat. Then, in [Sch84],Schoen completed the proof that a solution to the Yamabe problem exists by usingthe positive mass theorem in the remaining cases. Namely, assume that (M, g) islocally conformally flat or n {3, 4, 5}. Let

Lg =4(n 1)n 2

g + Scalg

1

be the conformal Laplacian of the metric g and P M . There exists a smoothfunction , the so-called Greens function of Lg, which is defined on M {P} suchthat Lg = P in the sense of distributions (see for example [LP87] ). Moreover,if we let r = dg(., P ), then in conformal normal coordinates has the followingexpansion at P :

(x) =1

4(n 1)n1 rn2+A+ (x) n1 = vol(Sn1)

where A R. In addition, is a function defined on a neighborhood of P and(0) = 0. On this neighborhood of P , the function is smooth if (M, g) is locallyconformally flat, and it is a Lipschitz function for n = 3, 4, 5. Hence, in both cases = O(r). Schoen has shown in [Sch84] that the positivity of A would imply thesolution of the Yamabe problem. He also proved that A is a positive multiple of themass of the asymptotically flat manifold (M,

4n2 g). Hence, in these special cases

the solution of the Yamabe problem follows from the positive mass theorem, whichwas proven by Schoen and Yau in [SY79, SY88].

In our article, we will give a short proof for the positivity of the constant termA in the development of the Greens function in case that M is spin and locallyconformally flat. The statement of this paper is weaker than the results of Witten[Wit81] and Schoen and Yau [Sch89, SY79]. The proof in our paper is inspiredby Wittens reasoning, but we have considerably simplified many of the analyticarguments. Wittens argument is based on the construction of a test spinor onthe stereographic blowup which is both harmonic and asymptotically constant. Weshow that the Greens function for the Dirac operator onM can be used to constructsuch a test spinor. In this way, we obtain a very short solution of the Yamabeproblem using only elementary and well known facts from analysis on compactmanifolds.

The last section shows how to adapt our proof to arbitrary spin manifolds ofdimensions 3, 4 and 5. In dimension 3 the proof is completely analogous. However,in dimensions 4 and 5, additional estimates have to be derived in order to getsufficient control on the Greens function of the Dirac operator.

2 The locally conformally flat caseIn this section, we will assume that (M, g) is a compact, connected, locally con-formally flat spin manifold of dimension n 3. The Dirac operator is denotedby D. A spinor is called D-harmonic if D 0. As the solution of the Yamabeproblem in the case of non-negative Yamabe invariant follows from [Tru68] we willassume that the Yamabe invariant is positive. Hence, the conformal class contains apositive scalar curvature metric. As dim kerD is conformally invariant, we see thatdim kerD = 0. We fix a point P M . We can assume that g is flat in a small ballBP () of radius about P , and that is smaller than the injectivity radius. Let(x1, . . . , xn) denote local coordinates on BP (). On BP () we trivialize the spinorbundle via parallel transport.

LEMMA 2.1. Let 0 PM . Then there is a D-harmonic spinor on M \ {P}satisfying

|BP () =xrn

0 + (x)

where (x) is a smooth spinor on BP ().

It is not hard to see, that in the sense of distributions

D = n1P0,

2

where P is the -function centered at P . Hence, by definition, 1n1 is the Greensfunction of the Dirac operator.

Proof. Our construction of follows the construction of the Greens function Gof the Laplacian in [LP87, Lemma 6.4]. Namely, we take a cut-off function withsupport in BP () which is equal to 1 on BP (/2). We set = 1rn1

xr 0 where

0 is constant. The spinor is D-harmonic on BP (/2) \ {P}. Outside BP () weextend by zero, and we obtain a smooth spinor on M \ {P}. As D|BP (/2) 0,we see that D extends to a smooth spinor on M . Using the selfadjointness ofD together with kerD = {0} we know that there is a smooth spinor 1 such thatD1 = D. Obviously, = + 1 is a spinor as claimed. 2

We now show that the existence of implies the positivity of A.

THEOREM 2.2. Let (M, g) be a compact connected locally conformally flat mani-fold of dimension n 3. Then, the mass A of (M, g) satisfies A 0. Furthermore,equality holds if and only if (M, g) is conformally equivalent to the standard sphere(Sn, can).

Proof. Let be given by lemma 2.1. Without loss of generality, we may assumethat |0| = 1. Let be the Greens function for Lg, and G = 4(n 1)n1. Usingthe maximum principle, it is easy to see that G is positive [LP87, Lemma 6.1]. Weset

g = G4

n2 g.

Using the transformation formula for Scal under conformal changes, we obtainScaleg = 0. We can identify spinors on (M \ {P},g) with spinors (M \ {P},g) suchthat the fiber wise scalar product on spinors is preserved [Hit74, Hij86]. Because ofthe formula for the conformal change of Dirac operators, the spinor

:= Gn1n2

is a D-harmonic spinor on (M \ {P},g), i.e. if we write D for the Dirac operator inthe metric g, we have D = 0. By the Schrodinger-Lichnerowicz formula we have

0 = D2 = +Scaleg

4 = .

Integration over M \BP (), > 0 and integration by parts yields

0 =

M\BP ()(, )dveg =

M\BP ()||2dveg

SP ()(e, )dseg.

Here SP () denotes the boundary BP (), is the unit normal vector on SP ()with respect to g pointing into the ball, and dseg denotes the Riemannian volumeelement of SP (). Hence, we have proved that

M\BP ()||2dveg =

12

SP ()e ||2dseg (2.3)

If is sufficiently small, we have

= G2

n2r

= (2 + o(2)

) r

(2.4)

dseg = G2(n1)

n2 dsg = G2(n1)

n2 n1ds =((n1) + o((n1))

)ds (2.5)

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where ds stands for the volume element of (Sn1, can), and

||2 = G2n1n2 ||2 =

(1

rn2+ 4(n 1)n1A+ r1(r)

)2 n1n2 1

rn1xr

0 + (x)2

where 1 is a smooth function. This gives

||2 = (1 + 4(n 1)n1Arn2 + rn11(r))2n1n2

(1 + 2rn1Re() + r2(n1)|(x)|2)

Noting that r(xr0) = 0, we get that on SP () and for small,

r||2 = 8(n 1)2n1An3 + o(n3) (2.6)

Plugging (2.4), (2.5) and (2.6) into (2.3), we get that for small

0

M\BP ()||2dveg = 4(n 1)2n1A

Sn1ds+ o(1) (2.7)

= 4(n 1)22n1A+ o(1) (2.8)

This implies that A 0.Now, we assume that A = 0. Then it follows from (2.7) that = 0 on M \{P}

and hence, is parallel. Since the choice of 0 is arbitrary, we obtain in this way abasis of parallel spinors on (M \ {P}, g). This implies that (M \ {P}, g) is flat andhence isometric to euclidean space. Let I : (M \{P}, g) (Rn, can) be an isometry.We define f(x) = 1 + I(x)2/4, x M . Then M \ {P}, f2g = f2G

4n2 g is

isometric to (Sn \{N}, can). The function f2G4

n2 is smooth on M \{P} and canbe extended continuously to a positive function on M . Hence, M is conformal to(Sn, can).

3 The case of dimensions 3, 4 and 5Now under the assumption that the dimension of M is 3, 4 or 5 we show how toadapt the proof from the last section to the case in which M is not conformally flat.For simplicity, in this article, we work with the synchronous trivialization, but thesame conclusions work with the Gauduchon-Bourguignon-trivialization [AHM03].Let us assume that (M, g) is an arbitrary connected spin manifold of dimensionn {3, 4, 5}. We choose any P M . After possibly replacing g by a metricconformal to g, we may assume that Ricg(P ) = 0. Let = (x1, , xn) be a systemof normal coordinates defined on U and centered at P . As before, let be theGreens function of the conformal Laplacian Lg, G = 4(n 1)n1.

We trivialize TM and M by synchronous frames ei and i, i.e. by frames whichare parallel along radial geodesics. We assume ei(P ) = /xi(P ). Let us introducea convenient notation for sections in this trivialization. For v =

viei(P ) : U

TPM = Rn we define v : U TM , v(q) =vi(q)ei(q), i.e. v is the coordinate

presentation for v. Similarly, for : U PM = Rn, =ii(P ) we write

(q) =i(q)i(q). In this notation, x is the outward radial vector field whose

length is the radius. Similarly, we write D for the Dirac operator on flat Rn and Dfor the Dirac operator on (M, g).

PROPOSITION 3.1. Let (M, g) be a compact connected spin manifold of dimensionn {3, 4, 5}. Let P M and 0 PM , then there is a spinor (0) which is D-harmonic on M \ P , and which has in the synchronous trivialization defined above

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the following expansion at P :

(0) =xrn

0 + 1(x) if n = 3

(0) =xrn

0 + 2(x) if n = 4

(0) =xrn

0 + (x) + 3(x) if n = 5

where 1 C0,a(Rn) for all a ]0, 1[, where, for all > 0, r2, r3, r1+|1|,r1+|2| and r1+|3| are continuous on Rn and where is homogeneous oforder 1 and even near P .

Remark. As before, the spinor 1n1(0) is the Greens function for the Diracoperator on M . The expansion of (0) could be improved but the statement ofproposition 3.1 is sufficient to adapt the proof of theorem 2.2.

Proof of positive mass theorem. Using this proposition, the proof of theorem2.2 can easily be adapted with = (0). As one can check, equation (2.7) is stillavailable in dimensions 3, 4 and 5. This is easy to see in dimensions 3 and 4. Indimension 5, we note that since is even near P and since xr is an odd vector field,we have

Re

Sn1

r

() ds =

Sn1Re() ds = 0

Equation (2.7) easily follows. This proves the positive mass theorem 2.2. 2

We will now prove Proposition 3.1.

Definition. Let ((Rn \ {0})) be a smooth spinor defined on Rn \ {0}. Fork R, we say that is homogeneous of order k if (sx) = sk(x) for all x Rn\{0}and all s > 0. This is equivalent to r = k.

PROPOSITION 3.2. Let be a spinor homogeneous of order k (n,1). Thenthere is a spinor , homogeneous of order k + 1, such that D() = .

Proof. Let be a homogeneous spinor of order k. Recall that D := xn1rn isthe Greens function for the Dirac operator on Rn. We define := D , i.e.

(x) =1

n1lim0

Rn\(Bx()B0())

x y|x y|n

(y) dy. x 6= 0

The integral converges for |y| as k < 1. The limit for 0 exists ask > n. Similarly one sees that is smooth, and one calculates D() = . Asimple change of variables shows that is homogeneous of order k + 1.

LEMMA 3.3 (Regularity Lemma). Let be a smooth spinor on Rn \ {0}. Weassume that D() = O(1r ) and iD() = O(

1r2 ) as r 0. Then, for all > 0, r

and r1+|| extend continuously to Rn.

Proof. As the statement is local, we can assume for simplicity that vanishesoutside a ball B0(R) about 0. Since D() Lq(Rn) for all q < n, from regularitytheory we get that Hq1 (Rn) for all q < n. The Sobolev embedding theoremthen implies that Lq(Rn) for all q > 1. Moreover, we have

|D(r)| r1|| +O(r1)

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Using Holders inequality, we see that D(r) Lq(Rn) for some q > n. Byregularity theory, we have r Hq1 (Rn) and by the Sobolev embedding theorem,r C0(Rn). This proves the first part of lemma 3.3. For the second part, weapply the same argument twice: a calculation on Rn \ {0} yields:

|D(r1+i)| (1 + )r|i| +O(r1) (3.4)

In the same way, we have:

|D(ri)| r1|i| +O(r2) (3.5)

For all i = 1, . . . , n we have D(i) = iD = O(r2) and D(i) Lq(Rn) for allq < n2 . Using the regularity theory and then the Sobolev embedding theorem, we getthat i Hq1 (Rn) for all q n2 , close to

n2 , such that r

1|i| Lq(Rn).Together with (3.5), this shows that D(ri) Lq(Rn) for some q > n2 . By theregularity and Sobolev theorems, we obtain that ri Lq(Rn) for some q > n.Now using (3.4), we obtain |D(r1+i)| Lq(Rn) for some q > n. Applyingregularity theory and the Sobolev theorems again, we get that r1+i C0(Rn).This proves that r1+|| C0(Rn). 2

Proof of Proposition 3.1.For any spinor , standard formulas in normal coordinates yield

D() = D +14

ijk

kijei ej ek (3.6)

with kij = g(eiej , ek) =

l12 x

lg(R(el, ei)ej , ek) + O(r2). Note that kij denotesthe Christoffel symbols of the second kind, which are, in general, not symmetric ini and j. The Bianchi identity implies that

i,j,ki6=j 6=k 6=i

kijei ej ek O(r2).

Hence,

ijk

kijei ej ek =

l

xlRiclk(P )ek +O(r2) = O(r2) (3.7)

Let be a cut-off function equal to 1 in a neighborhood V of P = 0 in M , andsupported in the normal neighborhood U . Let 0 be a constant spinor on Rn. Wedefine on U \ {0} by

=

rn1xr

0

and extend it with zero on M \U . Then, is smooth on M \{P} and is D-harmonicnear P (see the locally conformally flat case) and by (3.6), near P we have

D() =1

4rn1

ijk

kijei ej ek xr

0.

Writing the Taylor development of

ijk kijei ej ek to order 3 at 0, we see by

(3.7) that we can write D() as a sum of a spinor which is homogeneous oforder 3 n and a spinor , smooth on U \ {0}, which satisfies = 0(r4n) and|| = 0(r3n).

D() = +

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If n = 3, + Lq(U) for all q > 1. Let be a cut-off function as above, then( + ) can be viewed as a spinor on Rn \ {0}, and := D (( + )) is asmooth spinor on Rn \ {0} such that D() = + near 0. By regularity theoryand the Sobolev embedding theorem, we get that Hq1 (Rn) for all q > 1, andhence C0,a(Rn) for all a (0, 1). Lemma 3.3 implies r1+|| C0(Rn).

If n = 4, we have + = 0(1r ) and |(+ )| = 0( 1r2 ). As in dimension 3, we can

find , a smooth spinor on Rn\{0} such thatD() = + near 0. We fix (0, 1).By the regularity lemma 3.3 we get r C0(Rn) and r1+|| C0(Rn).

If n = 5, by proposition 3.2, we can find a spinor homogeneous of order 1 suchthat D() = . Moreover, = 0(1r ) and |

| = 0( 1r2 ). Proceeding as in dimen-sion 3 and using lemma 3.3, we can find a spinor f smooth on Rn \ {0} such thatD(f) = near 0, such that rf C0(Rn) and such that r1+|f | C0(Rn). Weset := (+ f).

Now, for all dimensions, we set

=

By (3.6), we have on V

D = D() D 14

ijk

kijei ej ek

Using (3.7) and the fact that D() D = 0, we get that D = O(r) and henceis of class C(M \ {P}) C0,1(M). As a consequence, there exists (M)of class C(M \ {P}) C1,a(M), a (0, 1) such that D = D. We now set(0) = . Proposition 3.1 follows. 2

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inequality Preprint 2004

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[Tru68] N.S. Trudinger. Remarks concerning the conformal deformation of Rie-mannian structures on compact manifolds. Ann. Sc. Norm. Super. Pisa,Sci. Fis. Mat., III. Ser., 22 265274, 1968.

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Authors address:

Bernd Ammann and Emmanuel Humbert,Institut Elie Cartan BP 239Universite de Nancy 154506 Vandoeuvre-les -Nancy CedexFrance

E-Mail: ammann at iecn.u-nancy.fr, humbert at iecn.u-nancy.fr

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